Suppose that $\alpha: V \rightarrow W$ is a linear map of finite-dimensional complex vector spaces. What is the dual map $\alpha^{*}$ of the dual vector spaces?

Suppose that we choose bases of $V, W$ and take the corresponding dual bases of the dual vector spaces. What is the relation between the matrices that represent $\alpha$ and $\alpha^{*}$ with respect to these bases? Justify your answer.